(More precisely, we use data splitting for training and estimating VI as detailed in alg:cap.)
Essentially, the linearized approximation of the NN is linear in the gradient feature map \(x\mapsto \nabla _{\theta } h_{\theta }(x)|_{\theta _f}\) . In fact, this gradient feature map induces the Neural Tangent Kernel (NTK, [1]}): for any \(x, x^{\prime }\in \mathbb {R}^p\) ,
\(\text{ker}_{\theta _f}(x, x^{\prime }) := \langle \nabla _{\theta } h_{\theta }(x)|_{\theta _f},\nabla _{\theta } h_{\theta }(x^{\prime })|_{\theta _f}\rangle .\) 
